\(\int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx\) [1017]

Optimal result

Integrand size = 24, antiderivative size = 182 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {5958887 \sqrt {1-2 x} \sqrt {3+5 x}}{2048000}+\frac {541717 (1-2 x)^{3/2} \sqrt {3+5 x}}{614400}+\frac {49247 (1-2 x)^{5/2} \sqrt {3+5 x}}{153600}-\frac {4477 (1-2 x)^{7/2} \sqrt {3+5 x}}{5120}-\frac {407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {65547757 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2048000 \sqrt {10}} \] Output:

5958887/2048000*(1-2*x)^(1/2)*(3+5*x)^(1/2)+541717/614400*(1-2*x)^(3/2)*(3 
+5*x)^(1/2)+49247/153600*(1-2*x)^(5/2)*(3+5*x)^(1/2)-4477/5120*(1-2*x)^(7/ 
2)*(3+5*x)^(1/2)-407/960*(1-2*x)^(7/2)*(3+5*x)^(3/2)-37/240*(1-2*x)^(7/2)* 
(3+5*x)^(5/2)-3/70*(1-2*x)^(7/2)*(3+5*x)^(7/2)[phone]/20480000*arcsin(1/ 
11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-74704869+1497221625 x+3564415220 x^2-3365562400 x^3-11818608000 x^4-723200000 x^5+14924800000 x^6+9216000000 x^7\right )-1376502897 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{430080000 \sqrt {3+5 x}} \] Input:

Integrate[(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(5/2),x]
 

Output:

(10*Sqrt[1 - 2*x]*(-74704869 + 1497221625*x + 3564415220*x^2 - 3365562400* 
x^3 - 11818608000*x^4 - 723200000*x^5 + 14924800000*x^6 + 9216000000*x^7) 
- 1376502897*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(43008 
0000*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {90, 60, 60, 60, 60, 60, 60, 64, 223}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (1-2 x)^{5/2} (3 x+2) (5 x+3)^{5/2} \, dx\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {37}{20} \int (1-2 x)^{5/2} (5 x+3)^{5/2}dx-\frac {3}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {37}{20} \left (\frac {55}{24} \int (1-2 x)^{5/2} (5 x+3)^{3/2}dx-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {37}{20} \left (\frac {55}{24} \left (\frac {33}{20} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {37}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {37}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {37}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {37}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {37}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {37}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\)

Input:

Int[(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(5/2),x]
 

Output:

(-3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))/70 + (37*(-1/12*((1 - 2*x)^(7/2)*(3 + 
 5*x)^(5/2)) + (55*(-1/10*((1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)) + (33*(-1/8*(( 
1 - 2*x)^(7/2)*Sqrt[3 + 5*x]) + (11*(((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/15 + 
(11*(((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (33*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x 
])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/6))/16))/ 
20))/24))/20
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 64

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp 
[2/b   Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] 
 /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] 
 || PosQ[b])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.65

method	result	size
risch	\(-\frac {\left (1843200000 x^{6}[phone] x^{5}-1272064000 x^{4}-1600483200 x^{3}[phone] x^{2}[phone] x -24901623\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{43008000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {65547757 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{40960000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(118\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (36864000000 \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] x^{5} \sqrt {-10 x^{2}-x +3}-25441280000 x^{4} \sqrt {-10 x^{2}-x +3}-32009664000 x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-498032460 \sqrt {-10 x^{2}-x +3}\right )}{860160000 \sqrt {-10 x^{2}-x +3}}\)	\(155\)

Input:

int((1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/43008000*(1843200000*x^6+1879040000*x^5-1272064000*x^4-1600483200*x^3+2 
87177440*x^2+540576580*x-24901623)*(-1+2*x)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5* 
x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)[phone]/40960000*10^(1/2) 
*arcsin(20/11*x+1/11)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.48 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {1}{43008000} \, {\left (1843200000 \, x^{6} + 1879040000 \, x^{5} - 1272064000 \, x^{4} - 1600483200 \, x^{3} + 287177440 \, x^{2} + 540576580 \, x - 24901623\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {65547757}{40960000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:

integrate((1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(5/2),x, algorithm="fricas")
 

Output:

1/43008000*(1843200000*x^6 + 1879040000*x^5 - 1272064000*x^4 - 1600483200* 
x^3 + 287177440*x^2 + 540576580*x - 24901623)*sqrt(5*x + 3)*sqrt(-2*x + 1) 
 - 65547757/40960000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3 
)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

Sympy [F(-1)]

Timed out. \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx=\text {Timed out} \] Input:

integrate((1-2*x)**(5/2)*(2+3*x)*(3+5*x)**(5/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.62 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx=-\frac {3}{70} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {37}{120} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {37}{2400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {4477}{3840} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {4477}{76800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {541717}{102400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {65547757}{40960000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {541717}{2048000} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

integrate((1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(5/2),x, algorithm="maxima")
 

Output:

-3/70*(-10*x^2 - x + 3)^(7/2) + 37/120*(-10*x^2 - x + 3)^(5/2)*x + 37/2400 
*(-10*x^2 - x + 3)^(5/2) + 4477/3840*(-10*x^2 - x + 3)^(3/2)*x + 4477/7680 
0*(-10*x^2 - x + 3)^(3/2) + 541717/102400*sqrt(-10*x^2 - x + 3)*x - 655477 
57/40960000*sqrt(10)*arcsin(-20/11*x - 1/11) + 541717/2048000*sqrt(-10*x^2 
 - x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (131) = 262\).

Time = 0.19 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.45 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(5/2),x, algorithm="giac")
 

Output:

1/3584000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 443)*(5*x + 3) + 94933)* 
(5*x + 3) - 7838433)*(5*x + 3) + 98794353)*(5*x + 3) - 1568443065)*(5*x + 
3) + 8438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*arcsi 
n(1/11*sqrt(22)*sqrt(5*x + 3))) + 11/192000000*sqrt(5)*(2*(4*(8*(4*(16*(10 
0*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 15385695)*(5 
*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcs 
in(1/11*sqrt(22)*sqrt(5*x + 3))) + 19/192000000*sqrt(5)*(2*(4*(8*(12*(80*x 
 - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5 
*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 
 3))) - 1091/9600000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x 
+ 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*s 
qrt(22)*sqrt(5*x + 3))) - 111/40000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 
1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sq 
rt(5*x + 3))) + 27/400*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5 
) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 27/25*sqrt(5)*(11*s 
qrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 
5))
 

Mupad [F(-1)]

Timed out. \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2} \,d x \] Input:

int((1 - 2*x)^(5/2)*(3*x + 2)*(5*x + 3)^(5/2),x)
 

Output:

int((1 - 2*x)^(5/2)*(3*x + 2)*(5*x + 3)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.73 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx=-\frac {65547757 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{20480000}+\frac {300 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{6}}{7}+\frac {1835 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}}{42}-\frac {4969 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{168}-\frac {166717 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{4480}+\frac {1794859 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{268800}+\frac {27028829 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{2150400}-\frac {8300541 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{14336000} \] Input:

int((1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(5/2),x)
 

Output:

( - 1376502897*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 184320 
00000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**6 + 18790400000*sqrt(5*x + 3)*sqrt 
( - 2*x + 1)*x**5 - 12720640000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 1600 
4832000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 2871774400*sqrt(5*x + 3)*sqr 
t( - 2*x + 1)*x**2 + 5405765800*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 2490162 
30*sqrt(5*x + 3)*sqrt( - 2*x + 1))/430080000